Lebesgue Integration on Euclidean Space

Transcription

1 Lebesgue Integration on Euclidean Space Frank Jones Department of Mathematics Rice University Houston, Texas Jones and Bartlett Publishers Boston London

2 Preface Bibliography Acknowledgments ix xi xiii 1 Introduction to R n 1 A Sets 1 B Countable Sets 4 C Topology 5 D Compact Sets 10 E Continuity 15 F The Distance Function 20 2 Lebesgue Measure one" 25 A Construction 25 B Properties of Lebesgue Measure 49 C Appendix: Proof of PI and P Invariance of Lebesgue Measure 65 A Some Linear Algebra 66 B Translation and Dilation 71 C Orthogonal Matrices 73 D The General Matrix 75 4h Some Interesting Sets 81 A A Nonmeasurable Set 81 B A Bevy of Cantor Sets 83 C The Lebesgue Function 86 D Appendix: The Modulus of Continuity 95 of the Lebesgue Functions

3 vi O Algebras of Sets and Measurable Functions A Algebras and cr-algebras 103 B Borel Sets 107 C A Measurable Set which Is Not a Borel Set 110 D Measurable Functions 112 E Simple Functions 117 Integration 121 A Nonnegative Functions 121 B General Measurable Functions 130 C Almost Everywhere 135 D Integration Over Subsets of R 139 E Generalization: Measure Spaces 142 F Some Calculations 147 G Miscellany 152 ff Lebesgue Integral on R n 157 A Riemann Integral 157 B Linear Change of Variables 170 C Approximation of Functions in L D Continuity of Translation in L Fubini's Theorem for R n The Gamma Function 199 A Definition and Simple Properties 199 B Generalization 202 C The Measure of Balls 205 D Further Properties of the Gamma Function 209 E Stirling's Formula 212 F The Gamma Function on R LP Spaces 221 A Definition and Basic Inequalities 221 B Metrie Spaces and Normed Spaces 227 C Completeness of L p 231

4 vii D The Case p = oo 235 E Relations between L p Spaces 238 F Approximation by C c (R n ) 244 G Miscellaneous Problems 246 H The Case 0 < p < Products of Abstract Measures 255 A Products of a-algebras 255 B Monotone Classes 258 C Construction of the Product Measure 261 D The Fubini Theorem 268 E The Generalized Minkowski Inequality Convolutions 277 A Formal Properties 277 B Basic Inequalities 280 C Approximate Identities Fourier Transform onl n 293 A Fourier Transform of Functions in L 1 (R n ) 293 B The Inversion Theorem 308 C The Schwartz Class 320 D The Fourier-Plancherel Transform 323 E Hubert Space 334 F Formal Application to Differential Equations 339 G Bessel Functions 344 H Special Results for n = l 352 I Hermite Polynomials Fourier Series in One Variable 367 A Periodic Functions 367 B Trigonometrie Series 373 C Fourier Coefncients 392 D Convergence of Fourier Series 400 E Summability of Fourier Series 410 F A Counterexample 418

5 viii G Parseval's Identity 421 H Poisson Summation Formula 428 I A Special Class of Sine Series 436 II ö Differentiation 447 A The Vitali Covering Theorem 448 B The Hardy-Littlewood Maximal Function 450 C Lebesgue's Differentiation Theorem 456 D The Lebesgue Set of a Function 458 E Points of Density 463 F Applications 466 G The Vitali Covering Theorem (Again).478 H The Besicovitch Covering Theorem 482 I The Lebesgue Set of Order p 491 J Change of Variables 494 K Noninvertible Mappings Differentiation for Functions onm 511 A Monotone Functions 511 B Jump Functions 521 C Another Theorem of Fubini 527 D Bounded Variation 530 E Absolute Continuity 544 F Further Discussion of Absolute Continuity 553 G Are Length 563 H Nowhere Differentiable Functions 570 I Convex Functions 576 Index 581 Symbol Index 587